Beyond the Local Void: A data-driven search for the origins of the Amaterasu particle

Technical Summary: Beyond the Local Void: A data-driven search for the origins of the Amaterasu particle

Problem Statement
The origins of Ultra-High-Energy Cosmic Rays (UHECRs), particularly those exceeding $10^{20}$ eV, remain unidentified. While previous efforts have linked UHECRs to catalogs of starburst galaxies (SBGs) and active galactic nuclei (AGN), these models struggle to explain events at the highest energies. The "Amaterasu" particle, detected by the Telescope Array with an energy of $244 \pm 29$ (stat.) $^{+51}_{-76}$ (syst.) EeV, presents a specific challenge: its arrival direction points toward the Local Void, a low-density region with no known active sources. Previous analyses using 1D backtracking or simple deflection estimates suggest the particle does not originate from known active galaxies, leading to hypotheses involving exotic physics (e.g., Lorentz invariance violation, superheavy dark matter) or transient sources. A rigorous, data-driven method is required to map the volume of the Universe consistent with such extreme events while accounting for complex propagation effects, magnetic deflections, and reconstruction uncertainties.

Methodology
The authors introduce a simulation-based inference framework utilizing Approximate Bayesian Computation (ABC) to constrain the origins of individual UHECRs. The method integrates realistic 3D propagation modeling with Bayesian parameter estimation, avoiding the need to explicitly define a likelihood function.

• Propagation Modeling: The framework employs CRPropa 3 to simulate UHECR propagation through 3D space. It includes all relevant interactions: photo-pion production, photo-disintegration, electron-pair production, nuclear decay, and adiabatic losses.
• Magnetic Fields:
  • Extragalactic Magnetic Field (EGMF): Modeled as a Gaussian random field with a Kolmogorov turbulence spectrum on a 25 kpc grid. Parameters include the root-mean-square field strength ($B_{rms}$) and coherence length ($L_c$), assigned log-uniform priors.
  • Galactic Magnetic Field (GMF): Modeled using the best-fit UF23 base model for the coherent field and JF12 Planck-tune components for turbulent fields.
• Parameter Space: The inference targets six free parameters: source position (Galactic longitude $l$, latitude $b$, and distance $D_{src}$), source energy ($E_{src}$), and EGMF properties ($B_{rms}$, $L_c$).
  • Source Composition: The particle is assumed to be an iron nucleus at the source to facilitate comparison with previous studies and to explore origins beyond the Local Void (requiring large deflections).
  • Priors: $E_{src}$ follows a power-law spectrum ($E^{-1}$). Source distances are uniformly sampled within a spherical volume up to $D_{max}$ (12 Mpc for nominal energy, 15 Mpc for low energy). The source direction prior is a von Mises-Fisher distribution centered on the backtracked direction of an iron nucleus, accounting for expected GMF and EGMF deflections.
• Inference Process (ABC):
  1. Parameters are sampled from the priors.
  2. $10^6$ particles are emitted isotropically from the sampled source location and propagated to Earth.
  3. Simulated events are compared to the observed Amaterasu data (arrival direction and reconstructed energy) using a distance metric.
  4. Parameter sets are accepted if the simulated event falls within $3\sigma$ of the observed energy and direction.
  5. Accepted sets are weighted based on the likelihood of the match, generating posterior distributions.
  6. Two energy scenarios are analyzed: a nominal case ($E_{det} = 244$ EeV) and a lower systematic bound ($E_{det} = 168$ EeV).

Key Results
Applying this framework to the Amaterasu particle yields the following findings:

• Source Candidates: The posterior distributions reveal a broader set of nearby source candidates than found in previous analyses. While the Local Void remains a region of interest, the results do not exclude sources within the Local Group or nearby galaxies.
• Mass Dependence: The accepted simulations are dominated by heavy nuclei (mass number $A > 28$). For the nominal energy case, 77.9% of accepted events correspond to heavy compositions, compared to 16.2% for medium and 5.9% for light.
• Distance Constraints:
  • For the nominal energy ($244$ EeV), the most probable source distance is $\sim 3$ Mpc. A secondary peak appears between 10–12 Mpc for heavy nuclei.
  • For the lower energy case ($168$ EeV), the most probable distance shifts to $\sim 4$ Mpc for light/medium masses, with a longer tail for heavy nuclei.
  • The arrival direction acts as the strictest selection criterion, limiting the effective distance horizon despite the large energy uncertainties.
• Deflection Analysis: The study quantifies that the Galactic Magnetic Field (GMF) dominates the deflection for lighter and intermediate masses, while the Extragalactic Magnetic Field (EGMF) contributes significantly only for heavy particles at larger distances or lower energies.
• Rigidity: The preferred source rigidity is found to be $R \approx 10^{19.3}$ V for the nominal energy and $R \approx 10^{18.85}$ V for the low energy case.

Significance and Claims
The paper claims to provide a foundational framework for "simulation-based analyses of cosmic rays at the highest energies." Its primary contribution is the ability to translate individual UHECR observations into directly interpretable source constraints by:

1. Jointly constraining energy and arrival direction, naturally incorporating their correlations and modeling uncertainties.
2. Marginalizing over uncertainties in magnetic fields, source composition, and reconstruction errors to produce 3D posterior distributions of possible source locations.
3. Challenging the "Local Void" exclusivity: By demonstrating that standard propagation models with heavy nuclei can produce consistent results for sources outside the Local Void, the authors suggest that exotic physics explanations may not be strictly necessary, though they do not rule them out.

The authors remain modest regarding the scope, noting that their current implementation is limited by computational constraints (e.g., the choice of a specific GMF model and a fixed source composition of iron) and that future work will aim to include a wider range of GMF models and treat particle type as a free parameter.